On the $$\Psi $$ Ψ -fractional integral and applications

Abstract

Motivated by the $$\Psi $$ -Riemann–Liouville $$(\Psi -\mathrm{RL})$$ fractional derivative and by the $$\Psi $$ -Hilfer $$(\Psi -\mathrm{H})$$ fractional derivative, we introduced a new fractional operator the so-called $$\Psi $$ -fractional integral. We present some important results by means of theorems and in particular, that the $$\Psi $$ -fractional integration operator is limited. In this sense, we discuss some examples, in particular, involving the Mittag–Leffler $$(\mathrm{M-L})$$ function, of paramount importance in the solution of population growth problem, as approached. On the other hand, we realize a brief discussion on the uniqueness of nonlinear $$\Psi $$ -fractional Volterra integral equation ( $$\mathrm{VIE}$$ ) using $$\beta $$ -distance functions.